Prime Graphs of Polynomials and Power Series Over Noncommutative Rings

The prime graph P G ( R ) of a ring R is a graph whose vertex set consists of all elements of R . Two elements x , y ∈ R are adjacent in the graph if and only if x R y = 0 or y R x = 0. An element a ∈ R is called a strong zero divisor in R if 〈 a 〉〈 b 〉 = 0 or 〈 b 〉〈 a 〉 = 0 for some nonzero element b ∈ R . The set of all strong zero divisors is denoted by S ( R ). In this paper, we study the prime graph of a ring R , considering S ( R ) as the set of vertices. In this way, we introduce the modified prime graph P G ∗ ( R ). We then investigate some combinatorial properties of the prime graphs P G ( R [ x ]) and P G ( R [[ x ]]) such as completeness, diameter, and girth, where R is a noncommutative ring.